Trigonometrically fitted nonlinear two-step methods for solving second order oscillatory IVPs

The construction of a class of nonlinear two-step methods for solving second order oscillatory IVPs is analyzed. These methods are exact for the linear space generated by the set of functions {sin(ωt),cos(ωt)} (trigonometrically fitted methods) but they do not require a previous knowledge of the frequency ω or a good approximation of it. Some trigonometrically fitted nonlinear two-step schemes with algebraic order up to four are derived and their stability and phase properties are analyzed. It is shown that the new schemes can be efficiently implemented by using a special vector operation (the vector product and quotient). The numerical experiments carried out show that the new nonlinear two-step schemes are more efficient than other standard and nonlinear methods proposed in the scientific literature for solving second order oscillatory differential systems.